The balance between acquisition and expenditure of energyis critical to the survival and reproductive success oforganisms. This balance depends on the interplay betweenenergy intake and processing, thermoregulation, metabolicexpenditures and production (Karasov, 1986). Nevertheless, assuggested by Hammond and Diamond (1997), the energeticsof an organism may also change as a result of its ownphysiological features and constraints. In endotherms, forexample, the elevated costs of thermoregulation require themto sustain expensive metabolic machinery, reflected by a highcost of maintenance. Metabolic rate, measured as oxygenconsumption (V˙O2), is a standard measure of these costs ofliving under various conditions. Metabolic rate measuredduring resting, post-absorptive adult and non-reproductiveanimals is known as basal metabolic rate (BMR) (McNab,1988). This variable measures the minimum cost ofmaintenance in euthermy, and has become one of the most

ubiquitous variables studied by physiological ecologists (seeMcNab, 2002). V̇O2 determined for an endotherm faced withconditions of extreme cold, usually accomplished by forcedconvection (Hayes and O’Connor, 1999) or a Helium–Oxygenatmosphere (Rosenmann and Morrison, 1974; Chappell andBachman, 1995), defines the thermoregulatory maximummetabolic rate (MMR). This variable has been shown tochange seasonally, as an adaptive feature in animals facingcold and fluctuating environments (Wickler, 1980; Hayes,1989; Holloway and Geiser, 2001), as well as among speciesinhabiting contrasting climates (Rosenmann and Morrison,1974; Bozinovic and Rosenmann, 1989). In eutherianmammals, a third respirometric variable, which is related tomaximum capacities, is non-shivering thermogenesis (NST).This variable is usually measured as the V˙O2 response tonorepinephrine injection, within thermoneutrality (Jansky,1973; Wunder and Gettinger, 1996), and is related to the

2145The Journal of Experimental Biology 206, 2145-2157© 2003 The Company of Biologists Ltddoi:10.1242/jeb.00396

Body size is one of the most important determinantsof energy metabolism in mammals. However, the usualphysiological variables measured to characterize energymetabolism and heat dissipation in endotherms arestrongly affected by thermal acclimation, and are alsocorrelated among themselves. In addition to choosing theappropriate measurement of body size, these problemscreate additional complications when analyzing therelationships among physiological variables such asbasal metabolism, non-shivering thermogenesis,thermoregulatory maximum metabolic rate and minimumthermal conductance, body size dependence, and the effectof thermal acclimation on them.

We measured these variables in Phyllotis darwini, amurid rodent from central Chile, under conditions ofwarm and cold acclimation. In addition to standardstatistical analyses to determine the effect of thermalacclimation on each variable and the body-mass-controlled correlation among them, we performed aStructural Equation Modeling analysis to evaluate the

effects of three different measurements of body size (bodymass, mb; body length, Lb and foot length, Lf) on energymetabolism and thermal conductance. We found thatthermal acclimation changed the correlation amongphysiological variables. Only cold-acclimated animalssupported our a priori path models, and mb appeared to bethe best descriptor of body size (compared with Lb and Lf)when dealing with energy metabolism and thermalconductance. However, while mb appeared to be thestrongest determinant of energy metabolism, there was animportant and significant contribution of L b (but not Lf) tothermal conductance. This study demonstrates howadditional information can be drawn from physiologicalecology and general organismal studies by applyingStructural Equation Modeling when multiple variablesare measured in the same individuals.

Key words: basal metabolic rate, maximum metabolic rate, thermalacclimation, structural equation modeling, body size, leaf-earedmouse, Phyllotis darwini.

Summary

Introduction

Body size as a latent variable in a structural equation model: thermalacclimation and energetics of the leaf-eared mouse

Roberto F. Nespolo*, Matías Arim and Francisco BozinovicCentro de estudios avanzados en Ecología y Biodiversidad, Departamento de Ecología, Facultad Ciencias

Biológicas, Pontificia Universidad Católica de Chile, PO Box 6513677, Santiago, Chile*Author for correspondence at present address: Instituto de Ecología y Evolución, Universidad Austral de Chile, Casilla 567, Valdivia, Chile

(e-mail: robertonespolo@uach.cl)

Accepted 24 March 2003

2146

formation of brown adipose tissue, a special thermogenic tissueidentified only in eutherians, that grows following chronic coldexposure (Himms-Hagen, 1990), and provides the explanationfor most of the seasonal variation seen in MMR (Bockler andHeldmaier, 1983; Wunder and Gettinger, 1996).

The roles of BMR, NST and MMR (known collectively as‘energy metabolism’) in maintaining the thermal homeostasisof endotherms depends on the efficiency of heat conservationin the body, which is, in turn, a function of body mass (mb)and thermal conductance (C). Thermal conductance, amongother things, is a function of the insulating properties of fur,the thermal gradient, evaporative water loss, the compositionof the atmosphere surrounding the body, and environmentalfactors such as radiant temperature and wind velocity (McNab,1980; Wooden and Walsberg, 2002). A reliable estimationof ‘wet’ thermal conductance (i.e. including evaporativewater loss; McNab, 1980) is obtained from the equationC=VO∑/(Tb–Ta), where Tb is body temperature and Ta isambient temperature (McNab, 1980). This simplifiedcalculation of C is useful as long as V̇O2 is recorded belowthermoneutrality (Anderson et al., 1997). It is known that C,as well as energy metabolism, can change seasonally and inresponse to thermal acclimation (Maddocks and Geiser, 2000;Merritt et al., 2001). The mechanisms and physiologicalprocesses responsible for these changes and their ecologicalsignificance are well understood (for reviews, see Jansky,1973; Heldmaier et al., 1985; Wunder and Gettinger, 1996;McNab, 2002).

Body mass, as a proxy for body size, is the main determinantof energy metabolism and thermal conductance in animals(Schmidt-Nielsen, 1995; Schleucher and Withers, 2001;McNab, 2002). This dependence is not simple, and complicatesstatistical analyses, especially when defining how to deal withbody mass when analyzing mass-independent physiologicaldata (Hayes, 1996, 2001; Christians, 1999). Several analyticalmethods exist for solving these problems, most of themrelated to common statistical procedures such as analysis ofcovariance (ANCOVA), multiple regression and residualanalysis (Christians, 1999). Body size, however, is anabstraction, whereas mb is a correlated variable that can bemeasured. It has been shown that other correlates of bodysize could yield different strengths of association withphysiological variables (Gosler, 2000; Tracy and Walsberg,2002), and hence different results from the analyses (Gosler,2000; Milner et al., 2000; Tracy and Walsberg, 2002). Theproblem is that most statistical methods are good at treatingeach physiological variable separately, but performing multi-variable analyses is not easy, especially when physiologicalvariables differ in their dependence on mb.

Structural equation modeling (SEM) is a powerful set ofprocedures which, combined with an adequate design, couldsolve these problems. With SEM it is possible to test completepath diagrams of causation and correlation among variables,and to include latent variables (variables that cannot bemeasured without error) or theoretical constructions associatedwith observed variables (Everitt, 1984; Cox and Wermuth,

1996). In addition, comparisons among standardized pathcoefficients are straightforward because they are weightedindices that represent the proportional contribution of eachcausation path to an observed, manifest variable (i.e. they canbe compared because they are scale-independent). SEM iswidely used as a research tool in psychology, sociology andmedicine (e.g. Koch et al., 2001; MacLullich et al., 2002), andhas been applied by evolutionary biologists (Crespi andBookstein, 1989), geneticists (Dohm, 2002) community(Wootton, 1993) and ecosystem ecologists (Ferguson, 2002)and, to a lesser extent, by physiological ecologists (Hayes andSchonkwiler, 1996).

This study had two aims: (1) to determine the mass-controlled effects of thermal acclimation on energy metabolism(BMR, NST and MMR) and thermal conductance (C) in amurid rodent Phyllotis darwini, and (2) to explore and comparethe effect of different measurements of body size (mb, Lt andLf) on energy metabolism and C, and their reliability as goodapproximations for body size using SEM. We hypothetized thatcold thermal acclimation has a profound effect overphysiological variables, modifying the effect of body size onthem. Accordingly, we predict that SEM models will adjustdifferently in warm- and cold-acclimated individuals.

Materials and methodsAnimals and thermal acclimation

Adult Phyllotis darwini Waterhouse 1837, 15 males(mb=64.6±9.1·g, mean ±S.D.) and 25 females (mb=57.7±11·g),were captured using 200 Sherman live traps between July andAugust 1999 (middle to late winter) in central Chile (33°29′S;70°56′W, 500·m above mean sea level. Animals weremaintained in the laboratory at ambient temperature (26±2°C,mean ± S.E.M.) and natural photoperiod. Individuals weremaintained with commercial rabbit and rat food pellets, andwater ad libitum. Each of 15 males were randomly assigned tothree non-pregnant females, one at a time, for 10 days each.Gravid females were isolated and weighed weekly using aSartorious (Gottingen, Germany) electronic balance (accurateto ±0.1·g). Young were isolated after weaning (at day 15)and maintained under the conditions described aboveuntil adulthood (approximately 150 days). Physiologicalmeasurements were performed on adult offspring on twooccasions (in 1 month): warm acclimation (30°C) followed bycold (12°C) acclimation. We obtained one large (N=108)dataset for each acclimation, which included 7 manifestvariables (i.e. measured variables: mb values recorded beforeBMR/NST, MMR and C measurements, plus the threerespirometric variables, plus C), and one smaller set of data(N=73) with two additional manifest variables: foot and bodylength (Lf and Lb, respectively), measured only in cold-acclimated animals.

Basal metabolic rate and non-shivering thermogenesis

Upon completing each thermal acclimation, and prior tomeasurements of BMR and NST, animals were fasted for 6·h

R. F. Nespolo, M. Arim and F. Bozinovic

2147Structural equation modeling and energetics in a mouse

(Nespolo et al., 2002). BMR and NST were measuredaccording to the following protocol. Oxygen consumption(VO∑) was measured in a computerized (Datacan V) open-flowrespirometry system (Sable Systems, Henderson, Nevada,USA). Animals were kept in steel metabolic chambers of1000·ml volume, at Ta of 30.0±0.5°C, which is within thethermoneutral zone for this species (Bozinovic andRosenmann, 1988). The metabolic chamber received dried airat a rate of 505±3·ml·min–1 from mass flow controllers (SierraInstruments, Monterey, CA, USA), which was enough toensure adequate mixing in the chamber. Air passed throughCO2-absorbent granules of Baralyme® and Drierite® beforeand after passing through the chamber, and was monitoredevery 5·s by an Applied Electrochemistry O2-analyzer, modelS-3A/I (Ametek, Pittsburgh, PA, USA). Oxygen consumptionvalues were calculated using equation·4a of Withers (1977).All metabolic trials were completed between 08:00 and16:00·h. Body mass was measured prior to the metabolicmeasurements using an electronic balance (to ±0.1·g), andrectal body temperature (Tb) was recorded at the end ofeach measurement with a Cu/copper-constant thermocouple(Cole–Parmer, Illinois, USA).

The experimental protocol was: (1) V˙O2 recorded for a 1.5·hperiod at rest, (2) intramuscular injection of norepinephrine(NE), followed by (3) a final 30·min period of V˙O2 recording.Recording ended when V̇O2 reached a sustained maximumvalue during a 10·min period. Doses of NE were calculatedaccording to Wunder and Gettinger’s equation (Wunder andGettinger, 1996: p. 133). Basal metabolic rate was estimatedas the lowest mean value recorded over a 3·min interval afterthe first period of V˙O2 recording. We previously measuredBMR to determine the optimal time of recording needed toreach minimum metabolism and we found that this speciesreaches a steady state after 15–20·min, with no changes of V̇O2

>15% in the following 3·h (see also Nespolo et al., 2002). Tocalculate NST from the recording, we used the highest sampleabove BMR, following the standard procedure (i.e. maximumV̇O2 after NE injection minus BMR; e.g. Klaus et al., 1988;Wunder and Gettinger, 1996).

Minimum thermal conductance

In addition to NST and BMR measurements, we measuredV̇O2 using the same procedure as above but at a Ta of 6±2°C,in order to determine ‘wet’ minimum thermal conductance (C).To monitor Ta inside the chamber, we used a copper-constantthermocouple set on its geometric center and approximately4·cm above the animal. Temperature was recorded every 4·swith a Data Logger (Digi-Sense, Illinois, USA). To avoid heatloss due to contact of the animal with the steel floor of thechamber, or with urine and feces, we covered the chamber floorwith 0.5·cm of sawdust, which was renewed before each newmeasurement. Rectal Tb was measured within 30·s of the lastrecording. The lowest value of V̇O2 within the last 10·minperiod of V̇O2 measurements was taken to determine C. Foreach individual we calculated minimum C using the equationC=VO∑/(Tb–Ta) (McNab, 1980).

Maximum metabolic rate

We measured MMR in a He–O2 atmosphere according tothe procedure of Rosenmann and Morrison (1974), using anopen circuit respirometer, as described by Chappel andBachman (1995). In brief, a mixture of He (80%) and O2 (20%)was passed through a volumetric flowmeter before enteringthe chamber (i.e. a positive pressure system), which wasmaintained at 1002±10·ml·min–1. This flow rate prevented thepartial oxygen pressure from falling below 20·kPa, a value farabove those considered hypoxic (Rosenmann and Morrison,1975). As in the case of BMR measurements, the mixturepassed through CO2-absorbent granules of Baralyme® andDrierite® before and after passing through the chamber, whichwas tightly sealed with Teflon® and Vaseline®. Chambertemperature (–5.0±0.5°C) and Tb were measured. The higheststeady-state 3·min period of recordings were taken as MMR.

Statistics and structural equation modeling

All statistical analyses were performed with Statisticaversion 6.0 (StatSoft, Tulsa, OK, USA). Differences inmeasured variables between acclimation groups (BMR, NST,MMR and C) were tested by repeated-measures (RM)ANCOVA with mb as a changing covariate (StatSoft). Partialproduct–moment correlations were used to test associationsbetween variables within each acclimation temperature,controlled by mb.

The Structural Equation Modeling (SEPATH) module ofStatistica (StatSoft) was used to compute standardized pathcoefficients among measured variables, and to test the overallpath diagram as a likely cause of observed data. We usedmaximum likelihood (ML) to estimate parameters, with theirrespective standard errors. Since this procedure is based onasymptotic statistics (large sample sizes), we performedMonte Carlo simulations to assess the behavior of our samplestatistic and the iteration procedure at different sample sizes.Our three measurements of body size were assumed asmeasured consequences of a latent variable, BODY SIZE,that we included in the model (see Appendix).Computationally, SEM makes a linear combination of thesevariables to build the latent variable (Shipley, 2000). Wehypothesized that energy metabolism and thermalconductance were primarily a function of the amount of tissuein the body, which is better described by mb, and not by linearmeasurements (Lb and Lf). For this reason, we included thepaths from body mass to physiological variables. The firstmodel (the ‘indirect’ model, Fig.·1) consisted of this causalstructure (i.e. mbBN→BMR, mbBN→NST; mbM→MMR;mbC→C) (see Fig.·1), which supposes that the effects of bodysize on physiological variables are expressed indirectlythrough body mass. The second model considered anadditional path structure from ‘BODY SIZE’ to eachmetabolic variable (the ‘direct–indirect’ model, Fig.·2).Hence, in this model the effect of body size overphysiological variables was decomposed into indirect effectsvia mb, as in the first model, and direct effects from ‘BODYSIZE’ to physiological variables (Fig.·2). In the context of the

2148

physiological variables studied here, ifthe indirect models were accepted, wewould conclude that mb is the variablethat best explains body size. Incontrast, the acceptance of thedirect–indirect model would mean thatboth Lf and Lb are important variablesexplaining body size, in addition to mb.

ResultsBody mass differed significantly

between acclimations (warm mb:57.3±12.5; cold mb: 65.6±11.3,t95=7.85, P<0.001). Thermalacclimation had a significant effect onall measured variables (BMR:F1,106=5.46, P=0.021; NST:F1,106=51.01, P<0.0001; MMR:F1,106=115.6, P<0.0001; C:F1,106=24.72, P<0.0001, RM-ANCOVA). Whereas metabolicvariables were higher in cold-acclimatedindividuals, thermal conductancepresented an opposite trend (Fig.·3).Correlations between all physiologicalvariables and mb were significant forboth acclimations (Fig.·4). Partialcorrelations (controlled by mb) showedthat in warm-acclimated animals, BMRwas significantly associated with MMRand with NST (Table·1). In cold-acclimated animals, there was only onesignificant partial correlation: BMRwith NST (Table·1). Morphologicalvariables measured in cold-acclimatedanimals were all highly intercorrelated(Table·2).

The adjustments of path models arepresented in Table·3. Based on nominalχ2 P-values, only three of eight pathmodels presented non-significantdifferences from the expectedcovariance structure (i.e. are supportedby the data). However, from the MonteCarlo simulated distribution, twoadditional models are supported by thedata (Table·3). In general, for the largedataset (i.e. using only mb as a measureof body size; see Materials andmethods), cold-acclimated individualspresented better adjustment than warm-acclimated ones (Table·3; models 1 and2), and for the small dataset (i.e. bodymass plus Lf and Lb, models 3–6, onlyin cold-acclimated animals; see

R. F. Nespolo, M. Arim and F. Bozinovic

Body

Siz e

BMR

NST

m bBN

m bM

m bC

MMR

C

u3

u

u

u

u

u

F L B L

u8u9

Bodysize

mbBN

mbM

mbC

2

u1

7

6

5

4

Lf Lb

B ody

Siz e

BMR

NST

m bBN

m bM

m bC

MMR

Cu3

u2

u1

u7

u6

u5

u4

FLu9 B Lu8

Bodysize

mbBN

mbM

mbC

Lf Lb

Fig.·1. Path diagram of the ‘indirect’ model tested with measured data. BMR, basalmetabolic rate; NST, non-shivering thermogenesis; MMR, maximum metabolic rate; C,thermal conductance; mbBN, body mass recorded at the moment of BMR and NSTmeasurement;mbM, body mass recorded at the moment of MMR measurement;mbC, bodymass recorded at the moment of C measurement; u, residual error. The broken line shows theadditional paths from variables other than mb, which were considered as indicators of bodysize: Lf (foot length) and Lb (body length). These variables were measured only for cold-acclimated individuals (the ‘small’ dataset). Latent variables are in circles and manifestvariables in rectangles. The small diagram at the upper left identifies the path model inTable·3 (see Results).

Fig.·2. Path diagram of the ‘direct–indirect’ model tested with observed data. Symbols andbroken lines as in Fig.·1.

2149Structural equation modeling and energetics in a mouse

Materials and methods), the single path Lf→Cimproved the adjustment significantly (Table·4). Thiseffect is also observed in the path diagram that includesonly indirect effects of body size to physiologicalvariables (model 3 versus4, Table·4), and in the pathdiagram including direct and indirect effects of bodysize to physiological variables (model 5 versus 6,Table·4). Comparisons between nested models(indirect model versusdirect–indirect model, seeTable·3) for warm- and cold-acclimated individualswere both significant (warm-acclimated animals:χ2[4]=15.44; P=0.004; cold-acclimated animals:χ2[4]=9.99; P=0.041).

Considering only the best-adjusted models (i.e. non-significant from Monte Carlo P-values, Table·3), pathdiagrams that included direct and indirect effects ofbody size over physiological variables were lessexplanatory than those that included only indirectpaths through mb (Figs·5–9). This is supported by thefact that in all cases all direct paths from body size arenon-significant, and several paths mb→‘physiologicalvariable’ have values above 1.0 (Figs·6, 7, 9), whichsuggest a poor adjustment. This allows us to discardall direct–indirect models (models 2, 5, 6 in Table·3,and Figs·6, 7 and 9), and leaves us only with model 1for cold-acclimated animals (Fig.·5) and model 4(Fig.·8).

Path coefficients of accepted models (Figs·5 and 8) aresimilar: large values in paths from body size to body massand smaller paths relating body mass to physiological

variables. A difference between both models is that in themore complex one (Fig.·8) the path relating mbC to C isconsiderably larger, and error paths for mbC and C are smaller(see Figs·5 and 8).

40 60 80 100

MM

R (

ml O

2 h–1

)

NS

T (

ml O

2 h–1

)

BM

R (

ml O

2 h–1

)

200

400

600

A

C

B

D

40 60 80 100

40

60

80

100

Body mass (g)

40 60 80 100

40

60

80

100

40 60 80 100

C (

ml O

2 h–1

deg

–1)

4

6

8

10

12

r2= 0.43*** r2 = 0.29***

r2= 0.22*** r2 =0.39***

r2 = 0.44*** r2 = 0.15***

r2 =0.27*** r2 =0.29***

Fig.·4. Correlations betweenbody mass and oxygenconsumption parameters.(A) MMR, maximummetabolic rate; (B) NST,non-shivering thermogenesis;(C) BMR, basal metabolicrate; (D) C, thermalconductance. Squaredcorrelation coefficients areshown for cold- (opensymbols, top) and warm-(closed symbols, bottom)acclimated individuals ineach case. ***, P<0.001.

Acclimation temperature (°C)M

etab

olic

rat

e (m

l O2 h

–1)

0

100

200

300

400

500

600BMRNSTMMR

15 30

The

rmal

con

duct

ance

(ml O

2 h

–1 d

eg–1

)

0

2

4

6

8

10

15 30

Fig.·3. Adjusted means (± 1S.E.M., N=108) of basal metabolic rate (BMR),non-shivering thermogenesis (NST), maximum metabolic rate (MMR) andthermal conductance in cold- (15°C) and warm- (30°C) acclimatedindividuals. Significant differences between acclimation temperatures werefound in all cases (ANCOVA, P<0.01; see Results).

2150

DiscussionAbsolute values of physiological variables and correlations

among them

Since this is not the first determination of physiological/energetic variables in P. darwini, it represents a goodopportunity to find out the extent by which ignoring the effectsof thermal acclimation on energy metabolism may have biasedprevious works. Reported values of BMR, MMR and C in P.darwini (Bozinovic and Rosenmann, 1988, 1989) fall exactlyin between our warm and cold measurements (in mass-specificunits, see Table·5). These same measurements made in anotherspecies of Phyllotis(P. xanthopygus), which inhabits highaltitudes, revealed very similar warm and cold values of BMRand NST, but MMR values were more extreme (Nespolo et al.,1999; Table·5). Predictions from allometric equations forrodents (Bozinovic, 1992) for BMR are 1.31·ml·O2·g–1·h–1 andfor MMR, 7.34·ml·O2·g–1·h–1. These values are closer to thecold-acclimated values in our study, but only the MMR values

fall inside the confidence intervals of our data (Table·5).Similarly, the expected allometric values for NST are2.90·ml·O2·g–1·h–1 and 5.56·ml·O2·g–1·h–1 for warm- and cold-acclimated animals, respectively (Wunder and Gettinger,1996). These values are above our measurements and falloutside our 95% confidence intervals for both acclimations(Table·5).

R. F. Nespolo, M. Arim and F. Bozinovic

Table·1.Partial correlations among energetic variables (controlled for by body mass)

MMR BMR NST C

MMR 1.0 0.247 (0.023) 0.198 (0.07) 0.127 (0.25)BMR –0.06 (0.58) 1.0 0.436 (0.001) 0.049 (0.66)NST 0.044 (0.66) 0.257 (0.009) 1.0 0.105 (0.34)C –0.096 (0.34) 0.065 (0.52) 0.023 (0.82) 1.0

MMR, maximum metabolic rate; BMR, basal metabolic rate; NST, non-shivering thermogenesis; C, thermal conductance.Values for warm- and cold-acclimated individuals are shown above and below the diagonal, respectively.P-values are in parentheses; significant comparisons at P<0.05 are in bold type.

Table·2.Correlation matrix for morphological variables incold-acclimated animals

Body mass Foot length Body length

Body mass 1.0 0.62 (0.0001) 0.78 (0.0001)Foot length 1.0 0.55 (0.0001)Body length 1.0

P-values are in parentheses; significant comparisons at P<0.05 arein bold type.

Bo dy

Size

BMR

m bBN

m bM

m bCu7

u6

u5

u4

0.94

4±0.

016*

0.924±0.019*

0.926±0.019*

0.146±0.035*

0.109±0.031*

0.143±0.035*

0.506±0.074*

0.564±0.068*

0.641±0.058*

0.626±0.060*

0.744±0.075*

0.682±0.076*

0.590±0.075*

0.608±0.075*

Cold-acclimated animals (14) = 19.32 (NSN= 108Boundaries=2%

Bodysize

NST

mbBN2

mbM

mbC

MMR

C

u3

u2

u1

χ )

Fig.·5. Path coefficients adjusted formodel 1, cold-acclimated animals(Table·3). Values are means ± 1S.E.M. Asterisks denote significantcoefficients (P<0.05). Latentvariables are in circles; manifestvariables are in rectangles.Adjustment statistics are fromTable·3 (only non-significant modelsare presented). Abbreviations as inFig.·1.

2151Structural equation modeling and energetics in a mouse

We found significant partial correlations between BMR andNST for both acclimations, and between BMR and MMR inwarm-acclimated animals only. Previous reports of (residual)intraspecific correlations between basal and maximummetabolic capacities show that, in general, they are small but

significant (Hayes and Garland, 1995). Since in endothermsBMR measures maintenance costs (i.e. operation ofmetabolically active organs), the coupling of BMR tomaximum capacities (in this case either MMR or NST) reflectsthe increase in systemic performance when maximumcapacities are higher. In cold-acclimated animals, however, theassociation BMR to MMR was absent (see also Bozinovic etal., 1990), which suggests that cold induced a disproportionateincrement of maximum performance, in agreement withsystemic adjustments.

Thermal acclimation

Thermal acclimation had an effect over all energeticvariables, which is well known for MMR and NST inmammals (Lynch, 1973; Wickler, 1980; Hayes and Chappell,1986; Merritt et al., 2001). Nevertheless, it is ratherparadoxical that C increased following cold acclimation sincepublished studies seem to suggest either no change underthese conditions, or a decrease in C (Maddocks and Geiser,

Table·3.Path models adjusted for warm and cold acclimated individuals

Boundary Nominal Monte Carlo Model Acclimation χ2 d.f. N rate (%) P-value P-value

(1) Warm 30.05 14 108 2 0.0075 <0.01

Cold 19.32 14 108 2 0.153 >0.05

(2)Warm 14.61 10 108 0.1 0.147 >0.05

Cold 9.33 10 108 0.1 0.501 >0.05

(3)Cold 50.12 27 71 0.1 0.0044 <0.01

(4)Cold 42.98 26 71 0.1 0.0194 >0.05

(5)

Cold 44.77 23 71 0.1 0.0042 <0.01

(6)

Cold 37.61 22 71 0.1 0.020 >0.05

The arrows and names were omitted for simplicity. Manifest (measured) variables are in rectangles and latent (non-measured) variables, i.e.body size, are in circles. Configurations are according to the path models presented in Figs 1 (models 1,3,4) and 2 (models 2,5,6).

Boundary rate is the proportion of parameter estimations that fell outside the parameter space in a Monte Carlo simulation with 100replications, using the observed sample size N for each case.

The nominal P-value corresponds to the probability level according to the theoretical χ2 distribution with the observed degrees of freedom(d.f.).

The Monte Carlo P-value is the observed probability from the χ2 simulated distribution.Significant comparisons at P<0.05 are in bold type.

Table·4.χ2 ratio tests for nested models showed in Table·3

Model

4 5 6

Model χ2 P χ2 P χ2 P

3 7.14[4] 0.0075 5.35[4] 0.253 12.51[5] 0.0284 1.79[3] 0.617 5.37[4] 0.2515 7.16[1] 0.0075

Values are χ2, with degrees of freedom in brackets, and thecorresponding P-value. Significant comparisons atP<0.05 are inbold type.

2152 R. F. Nespolo, M. Arim and F. Bozinovic

Bo dy

Size

BMR

NST

m bBN

m bM

m bC

MMR

u3

u2

u1

u7

u6

u5

u4

0.95

7±0.

017*

0.77

7±0.

459

–0.7

41±0

.236

*

–0.2

06±0

.264

–0.352±0.266

0.90

1±0.

025*

0.917±0.022*

0.160±0.040*

0.084±0.032*

0.188±0.044*

0.639±0.090*

0.850±0.072*

–0.138±0.455

0.582±0.092*

0.863±0.251* 0.689±0.085*

1.175±0.217*

0.563±0.253*

Bodysize

mbBN

mbM

mbC C

Warm-acclimated animals (10) = 14.61 (NSN= 108Boundaries=0.1%

2χ )

Fig.·6. Path coefficients adjusted formodel 2 (indirect effects of latent bodysize on mass), warm-acclimatedindividuals (see Table·3) Values aremeans ± 1 S.E.M. Asterisks denotesignificant coefficients (P<0.05).Latent variables are in circles; manifestvariables are in rectangles. Adjustmentstatistics are from Table·3 (only non-significant models are presented).Abbreviations as in Fig.·1.

Bod y

Size

BMR

NST

m bBN

m bM

m bCu3

u2

u1

u7

u6

u5

u4

0.92

1±0.

019*

0.35

5±0.

230

–0.6

54±0

.355

–0.4

75±0

.319

–0.468±0.251

0.94

3±0.

016*

0.928±0.018*

0.138±0.033*

0.152±0.035*

0.111±0.031*

0.696±0.082*

0.657±0.079*

0.314±0.225

0.570±0.075*

1.061±0.235* 0.578±0.077*

1.122±0.320*

1.012±0.304*

Bodysize

mbBN

mbM

mbC

MMR

C

Cold-acclimated animals (10) = 9.33 (NSN= 108Boundaries=0.1%

2χ )

Fig.·7. Path coefficients adjusted formodel 2 (indirect effects of latent bodysize on mass), cold-acclimatedindividuals (see Table·3). Values aremeans ± 1 S.E.M. Asterisks denotesignificant coefficients (P<0.05).Latent variables are in circles; manifestvariables are in rectangles. Adjustmentstatistics are from Table·3 (only non-significant models are presented).Abbreviations as in Fig.·1.

2153Structural equation modeling and energetics in a mouse

2000; Sharbaugh, 2001). This means that insulation is reducedin cold-acclimated individuals, which suggests a poorperformance under cold conditions. However, our measure ofC (‘wet’ thermal conductance; McNab, 1980) is computed

from V̇O2. This measurement has several useful properties; forexample in small bodies it has been shown to be a betterpredictor of heat loss than dry C (i.e. heat loss rate measuredin dead animals or carcasses) (Klaassen et al., 2002). Actually,

Bod y

Size

BMR

NST

m bBN

m bM

m bC

MMR

u3

u2

u1

u7

u6

u5

u4

0.96

7±0.

009*

0.996±0.005*

0.951±0.013*

0.008±0.010

0.066±0.018*

0.095±0.024*

0.519±0.090*

0.595±0.079*

0.580±0.082*

0.883±0.117*

0.731±0.093*

0.645±0.095*

0.664±0.095*

0.591±0.093*

F L B L

u8u9

0.60

4±0.

079* 0.778±0.049*

–0.416±0.138*

0.635±0.095* 0.395±0.076*

Bodysize

mbBN

mbM

mbC C

Lf Lb

Cold-acclimated animals (14) = 42.98 (NS)N= 71Boundaries=0.1%

2χ

Body

Size

BMR

NST

m bBN

m bM MMR

u3

u2

u1

u7

u6

u5

u4

0.99

5±0.

005

4.80

1±4.

929

–0.5

59±0

.405

–0.0

83±0

.385

–0.352±0.266

0.96

4±0.

010*

0.95

4±0.

012*

0.089±0.023*

0.010±0.010

0.072±0.019*

0.709±0.095*

0.645±0.095*

–4.197±4.931

0.441±0.270

0.799±0.320* 0.594±0.093*

1.057±0.393*

0.676±0.378

F Lu9

B Lu8

–0.4

26±0

.151

*

0.619±0.076* 0.781±0.048* 0.389±0.076*

m bC

Bodysize

mbBN

mbM

C

Lf Lb

0.617±0.094*

mbC

Cold-acclimated animals (10) = 37.61 (NS)N= 71Boundaries=0.1%

2χ

Fig.·8. Path coefficients adjustedfor model 1 (direct effects oflatent body size on mass), cold-acclimated individuals includingfoot length (Lf) and body length(Lb) as proxies for body size (seeTable·3). Values are means ± 1S.E.M. Asterisks denote significantcoefficients (P<0.05). Latentvariables are in circles; manifestvariables are in rectangles.Adjustment statistics are fromTable·3 (only non-significantmodels are presented).Abbreviations as in Fig.·1.

Fig.·9. Path coefficients adjusted formodel 2 (indirect effects of latent bodysize on mass), cold-acclimated individualsincluding foot length (Lf) and body length(Lb) as proxies for body size (see Table·3).Values are means ± 1S.E.M. Asterisksdenote significant coefficients (P<0.05).Latent variables are in circles; manifestvariables are in rectangles. Adjustmentstatistics are from Table·3 (only non-significant models are presented).Abbreviations as in Fig.·1.

2154

carcasses increase the surface-area-to-volume ratio (Klaassenet al., 2002) and dead animals lack piloerection and otherphysiological mechanisms that reduce heat loss in liveanimals (Turnpenny et al., 2000). So, wet C accounts forall the ways that heat loss occurs in live animals, includingthose that stem from increases in metabolic rate (higherrespiratory gas exchange), and enhanced peripheralcirculation due to increase in interscapular brown adiposetissue (Lynch, 1973). It is known that after cold acclimation,murids can increase blood flow to brown adipose tissueby a factor of ten (Puchalski et al., 1987). This change isenough to increase heat loss in peripheral areas of the thorax,which is demonstrated by infrared thermography (Jackson etal., 2001). Since our cold-acclimated animals showed anincrease in C (which reflect the rate of heat loss), all otherthings being equal, it would be reasonable to observe a smallincrease in C, which was the case (less than 10%; seeResults).

There still remains the question of why other studies havenot detected such an effect of thermal acclimation on C. Webelieve this is because our analysis is more powerful, due tothe control for body mass. In fact, when expressed as mass-specific units, both BMR and C show non-significant effectsof thermal acclimation (Table·5). This illustrates theconfounding effect of mb when analyzing mass-specific data.When this ratio is used, the residual variance is increasedbecause mb is measured with an error that is not controlled,because it is included as a denominator of another variable(VO∑) which, in turn, is also measured with an error. It followsthat using this procedure, the residual variance of the linearmodel must be enlarged (in the case of an ANOVA), with theconsequent loss of power in the overall analysis (Packard andBoardman, 1999; Hayes, 2001). The use of mass-specific unitshas probably obscured several subtle effects of acclimationand many other factors on the physiological variablespublished so far (e.g. Holloway and Geiser, 2001; Tracy andWalsberg, 2002). In agreement with previous authors (Hayes,1996, 2001; Packard and Boardman, 1999; Christians, 1999;Lleonart et al., 2000), this evidence suggests that mass-specific variables in the literature should be treated withcaution.

Structural Equation Modeling and body mass as a proxy ofbody size

Our three measurements of body size were highlyintercorrelated, reflecting the fact that all were reliableestimators of body size. Also, path coefficients relatinglatent body mass with mb, Lf and Lb were large and significantin all cases. It is interesting, however, that while mb

appears to be the strongest determinant of energymetabolism, there is an important and significant contributionof Lb (and not Lf) to C. This is not surprising since C is afunction of surface-area-to-volume, and Lb is the total lengthof the body, and hence is geometrically related to surfacearea.

In spite of the generalized use of SEM in biology, inaddition to the study of Hayes and Shonkwiler (1996), wewere unable to find other published works that have addressedthe problem of causation of physiological variables in smallendotherms. Nevertheless, SEM has been recognized as apowerful tool for correlational experiments by severalorganismal biologists. For example, Pigliucci and Schlichting(1998) applied path analysis to investigate the differencesamong populations and the plasticity of plant architecture infruits of Arabidopsis; Kause et al. (1999) used the fullcapabilities of SEM to test the effects of leaf quality on larvaltraits in six sawfly species; Miles et al. (2000) used SEM topredict survivorship from life history variables in a lizard; andFerguson (2002) used SEM to discriminate between direct andindirect effects of demographic and environmental variableson age at maturity in a moose. These examples, together withour results, illustrate how SEM could be a powerful tool todistinguish between contrasting causal models in organismalbiology.

SEM proved to be useful for our data analysis in the sensethat we could test specific models and subtle effects, such asthe inclusion of specific paths, to explore the change in theoverall goodness of fit. This enabled us to conclude that ourmodels were good explanations of the relationship betweenbody size and energetic variables. However, this was only truefor cold-acclimated animals; warm-acclimated animals didnot adjust to our models, possibly because NST and MMRonly have physiological significance in cold-acclimated

R. F. Nespolo, M. Arim and F. Bozinovic

Table·5. Values of energy metabolism (see Table·1) and thermal conductance reanalyzed as mass-specific values for comparisionwith previous studies (see Discussion)

Acclimation

Warm Cold F1,107 P

BMR (ml·O2·g–1·h–1) 1.17 (1.11-1.22) 1.22 (1.16-1.28) 2.28 0.134NST (ml·O2·g–1·h–1) 2.68 (2.53-2.83 3.39 (3.25-3.54) 57.59 <0.0001MMR (ml·O2·g–1·h–1) 5.56 (5.37-5.76) 7.38 (7.18-7.58) 194.18 <0.0001C (ml·O2·g–1·h–1·deg–1) 0.141 (0.137-0.146) 0.141 (0.137-0.145) 0.326 0.570

BMR, basal metabolic rate; NST, non-shivering thermogenesis; MMR, maximum metabolic rate; C, thermal conductance.Values are means ± 95% confidence intervals.Significant differences between acclimations are in bold type (P<0.05).

2155Structural equation modeling and energetics in a mouse

individuals. This raises the importance of thermal acclimationwhen measuring respirometric variables in endotherms.Likewise, the partial correlation that was detected amongphysiological variables was not detected by the SEM analysis(when we included such paths, the iteration did not convergebecause of the appearance of singular matrix). This occursbecause SEM models can often be structurally identified, butnumerically underidentified (Shipley, 2000, p. 167). These arelimitations of SEM that cannot be ignored, and make itespecially important to complement the SEM procedure withstandard statistical analyses.

An interesting outcome from the SEM analysis is that thesingle path Lb→C significantly improved the goodness of fit.Our best models were those that related body size withphysiological variables indirectly, through mb, and not bydirect paths. Moreover, by examining the accepted pathmodels, and comparing error paths, we could infer that thestrength of association between body size and morphologicaltraits is considerably larger than the (indirect) associationbetween body size and physiological variables. Similarly, itis clear that C has a higher (indirect) dependence from bodysize than energy metabolism. None of these conclusionswould have been possible using only standard statisticalprocedures.

List of symbols and abbreviationsLb body lengthLf foot length mb body massmbBN body mass at the moment of BMR/NST

determinationmbM body mass when MMR was measuredmbC when C was recordedV̇O2 oxygen consumptionTa ambient temperatureTb body temperatureBMR basal metabolic rateC thermal conductanceML maximum likelihoodMMR maximum metabolic rateNE norepinephrineNST non-shivering thermogenesis

AppendixStructural Equation Modeling procedure, Monte Carlo

simulations, minimum sample size estimation and hypothesistesting between models

The minimum number of individuals needed for ouranalyses was assessed by simulating different sample sizes in100 replications, and testing the number of boundary cases ineach simulation (an estimation outside the parameter space;e.g. an estimated correlation coefficient outside the range|0–1|). Usually, sample sizes above N=60 gave less than 5% ofboundary cases in the iteration for all models, which we

considered a reasonable rate. The significance of the overallpath model was assessed using the chi-squared (χ2) statisticcomputed from the departure of observed data from theexpected covariance matrix (Shipley, 2000). The P-value ofthis statistic was obtained (1) from the theoretical χ2

distribution at the corresponding degrees of freedom, and (2)from the simulated distribution obtained from the Monte Carloexperiment (Manly, 1998). The last procedure is consideredmore powerful since the sampling distribution does not alwaysapproach the theoretical one (i.e. χ2) with finite data (Manly,1998). The null hypothesis of a perfect fit means that the datasupport the proposed model (Shipley, 2000). By contrast, asignificant χ2 (P<0.05) means that the model is not supportedby the data and needs to be modified. Structural equationmodeling has two basic assumptions: multivariate normalityand linearity among variables. Our data satisfied bothassumptions and no transformation was necessary. Althoughphysiological variables scale non-linearly with mb (Schmidt-Nielsen, 1985), the best approximation when data have nomore than one order of magnitude is a linear approximation(see Results). We created a latent exogenous variable, ‘BODYSIZE’, which was considered the cause of the three followingmeasurements of body mass that we used: (1) body mass at themoment of BMR/NST determination (mbBN), when MMR wasmeasured (mbM), and when C was recorded (mbC). Usually,mbBN was lower than mbM and mbC because animals werefasted before the measurement of BMR/NST. We consideredbody size as the cause of the manifest variables mb, Lf and Lb

(the latter two only for the small dataset). However, not allof these variables were considered a direct cause of energymetabolism and thermal conductance.

Nested models (i.e. models differing in path number but withsame number of variables) were compared with χ2 ratio testssince the difference in the maximum likelihood χ2 valuesbetween nested models is, itself, asymptotically distributed asa χ2 distribution (Shipley, 2000). The degrees of freedom forthe resultant χ2 distribution are equal to the number ofparameters that have been freed in the nested model, which isthe same as the change in the degrees of freedom between thenested models (Shipley, 2000). This test allowed us to evaluatewhether different models that included the same variables werestatistically different, using the same set of data. To select thebest models, we proceeded as follows: first, we looked at thestatistics of the adjustment of the different models and selected(and presented) those that yielded a non-significant χ2. Second,we performed χ2 ratio tests to analyze the specific contributionof single paths to the overall model, between nested models.Third, path coefficients of selected models (i.e. non-significantfrom the Monte Carlo P-value) were examined, and those thatpresented more significant paths were judged to be the best.Although the effects of thermal acclimation were evaluatedindividually for each physiological variable, the condition ofmeasurement (cold- or warm-acclimated) is explicitly stated ineach path diagram.

This work was funded by Fondecyt grant 2000002 to R.N.

2156

and FONDAP grant 1051-0001 (program 1) to F.B. M.A.acknowledges a DIPUC fellowship.

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